Ceres and Vesta will be very close to each other over the next few days. (That is they appear to be close to each other when observed through a telescope).
What are Ceres and Vesta? Ceres is the largest asteroid and the first to be discovered, Vesta is the second largest asteroid and the fourth to be discovered. They are too dim to be seen naked eye, but bright enough to be easily seen in a small telescope (provided you know where to look). There is more information on this Sky and Telescope article....
Pluto reaches best visibility on July 4th, though it is possible to observe it for some time. Pluto is not visible naked eye, but if you know exactly where to look, it is possible to see it in many amateur telescopes (provided it is large enough or if the sky is dark enough). The trick is to know which of the many dots of light in the field are stars and which is Pluto. More information on this page....
When Ceres was discovered, it was considered to be a new planet. There was something called the "Titius-Bode Law", which predicted how far each planet should be from the sun. This relationship was reasonably accurate in describing the planets from Mercury to Uranus if you assumed there was a missing planet between Mars and Jupiter. (Neptune and Pluto had yet to be discovered). Ceres seemed to be that missing planet.
Over time when other objects were discovered between Mars and Jupiter, such as Pallas (an asteroid somewhat smaller than Vesta) and Vesta, they too were considered planets. However as more and more objects were found eventually it no longer made sense to consider any of them as planets; we now call them asteroids.
Since that time Neptune and Pluto were discovered and added to the list of planets. Eventually most scientists came to the conclusion that the Titius-Bode law was mere coincidence, and not a useful tool for predicting the existence of new planets. It did not correctly predict Neptune's distance from the sun, and it only predicted one object between Mars and Jupiter, not the multitude of objects now known.
Beginning twenty years or so ago, there was reason to question the status of Pluto. Objects beyond Pluto were discovered. Some are known as Kuiper-Belt objects after one of the astronomers who predicted their existence. Others are in a region called the "Scattered Disk." One of the later is bigger than Pluto. All these objects (along with Pluto and the so far unobserved Oort Cloud objects) are collectively called TNOs (trans-Neptunian objects). So it is still reasonable to consider Pluto as the ninth planet, or is it merely one of many TNOs?
Now some argued that objects in the solar system should be divided based on size. If so we have "large" objects that are in "hydrostatic equilibrium" (that is they are large enough that gravity forces them to be round). And we have smaller objects (which are generally lumpy-not round). All the traditional planets (including Pluto) would be "large". Also in this group would be Ceres, the largest moons of the solar system (including the Earth's Moon), and the largest TNOs. This list includes between 30 and 40 known objects and almost certainly includes dozens of objects yet to be discovered. This does not include the smaller asteroids (that is all asteroids except Ceres and Vesta), comets, the smaller moons and the smaller TNOs. Vesta is generally considered too small to be in hydrostatic equilibrium; however whether it is or is not is debatable.
It seemed that a clear definition of the word planet was needed. The word has always been poorly defined. With the discovery of a TNO bigger than Pluto, the issue could no longer be avoided. A definition of planet was constructed that included all of the traditional planets, with the notable exception of Pluto, and did not include any other object in hydrostatic equilibrium (moons, Ceres, the handful of large TNOs).
The definition essentially says a planet must be in hydrostatic equilibrium, and it must be the dominant gravitational object in its vicinity. This cuts off Pluto (which is under the influence of Neptune) and the other TNOs (which are under the gravitational influence of other TNOs). It cuts off Ceres (which is under the influence of Jupiter and other asteroids), and it cuts off moons (as each is under the influence of its parent body). Objects that are not planets would be called a moon if they were in a clear orbit around another object, or a dwarf-planet if they are large enough to be in hydrostatic equilibrium, not a moon, and didn't dominate their environment gravitationally. Both Pluto and Ceres are considered dwarf-planets. Vesta is probably not a dwarf-planet. This definition was adopted by the International Astronomical Union (IAU) in 2006.
An alternate argument for excluding Pluto goes as follows: Pluto is very different than rocky planets (such as Earth and Venus) and it is very different than the gas planets (such as Jupiter and Saturn). Pluto is best described as a "dirty snowball" and as such is very similar to Kuiper-Belt objects. In fact it makes sense to say that Pluto is a Kuiper-Belt object.
Note, not everyone is happy with the IAU definition...
For more reading, here are some Wikipedia articles on the topics in this post....
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